MRI ReconLab ShanghaiTech

Iterative SENSE

Iterative SENSE solves linear problem $E\hat{m}=b$ as a least square minization $\min_m|Em-b|_2^2$. Applying the gradient descent method, the iterative SENSE algorithm is given by solving $\min_m|(E^HE)m-E^Hb|_2^2$. Where $E=UFC$

Gradient Descent:

Initialize:

$$\begin{aligned} m_0&=E^Hb\\ r_0&=m_0-E^HEm_0\\ \end{aligned}$$

Iteration:

$$\begin{aligned} \alpha_k&=\cfrac{r_k^Tr_k}{r_K^TE^HEr_k}\\ m_{k+1}&=m_k+\alpha_k r_k\\ r_{k+1}&=r_k-\alpha_kE^HEr_k\\ \end{aligned}$$

Compressed Sensing

SALSA (split augmented Lagrangian shrinkage algorithm) algorithm is a fast and robust algorithm for solving formulations of a unconstrained optimization problem with a non-smooth regularization term. SALSA is based on variable splitting and augmented lagrangian methods.

The CS reconstruction problem is formulated as follows:

$$\begin{aligned} &\min\limits_{m,\theta} |Em-b|_2^2 + \lambda\Phi(\theta)\\ &\mathrm{s.t.}\quad m = \theta\\ \end{aligned} $$

Where $E=UF$ is the forward operator for image to k-space and undersampling, $b$ is the undersampled k-space data, $m$ is the image to be reconstructed, $\theta$ is the auxiliary variable introduced by the variable splitting method, $\Phi$ is the regularization term (Total Variation), and $\lambda$ is the regularization parameter.

The algorithm is given by the following steps:

Initialization:

$$\textit{choose}\quad \mu>0, m_0, \theta_0, d_0$$

$$\bar{b} = E^Hb$$

Iteration

$$\begin{aligned} m_k'&=\theta_k+d_k\\ r_k&=\bar{b}+\mu m_k'\\ m_{k+1}&=\frac{1}{\mu}(I-E^HE)r_k\\ \theta_k'&=m_{k+1}-d_k\\ \theta_{k+1}&=\Psi_{r\Phi/\mu}(\theta_k')\\ d_{k+1}&=d_k-\beta_{k+1}+\theta_{k+1}\\ \end{aligned}$$

Where $\Psi_{r\Phi/\mu}$ denotes shrinkage/thresholding function for denoiseing the image.

bughht/MRI_ReconLab_ShanghaiTech

MRI ReconLab ShanghaiTech

Iterative SENSE

Compressed Sensing